If |a|&gt;|a+b+c| prove that there is complex root such that |z|&lt;2.Let ax2+bx+c=0 be a quadratic equation where |a|&gt;|a+b+c|, a, b, c∈R. Prove that this equation has at least one solution z∈C such that |z|&lt;2.

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If |a|&amp;gt;|a+b+c| prove that there is complex root such that |z|&amp;lt;2.Let ax2+bx+c=0 be a quadratic equation where |a|&amp;gt;|a+b+c|, a, b, c∈R. Prove that this equation has at least one solution z∈C such that |z|&amp;lt;2.

despescarwh9 · Accepted Answer

If both roots s,t are real then we havea(x−s)(x−t)=ax2−a(s+t)x+∗=0. Now |a|&amp;gt;|a−a(s+t)+∗|⇒1&amp;gt;|1−s−t+st|=|(1−s)(1−t)|.Clearly |s| and |t| cannot both be greater than 2. The double root case is straightforward.If both roots s=u+iv and t=u−iv are complex we havea(x−(u+iv))(x−(u−iv))=ax2−2aux+a(u2+v2)=0.Now |a|&amp;gt;|a−2au+a(u2+v2)|⇒1&amp;gt;|1−2u+(u2+v2)|.If |s| (and hence |t|) are greater than 2 then u2+v2&amp;gt;4.

If |a|> |a+b+c| prove that there is complex root such

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